Rigidity of closed minimal hypersurface in $\mathbb{S}^5$

TL;DR

This paper proves that closed minimal hypersurfaces in a 5-sphere with constant second fundamental form length, constant 3-mean curvature, and a fixed number of principal curvatures are isoparametric, supporting the Chern Conjecture.

Contribution

It establishes a rigidity result for minimal hypersurfaces in spheres under specific curvature conditions, confirming cases of the Chern Conjecture.

Findings

Hypersurfaces with constant $H_3$ and $g$ are isoparametric.

The squared length of the second fundamental form $S$ can only be 0, 4, or 12.

Supports the validity of the Chern Conjecture in this setting.

Abstract

Let $M^4\to \mathbb{S}^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ in a $5$-dimensional sphere $\mathbb{S}^5$. In this paper, we prove that if $3$-mean curvature $H_3$ and the number $g$ of the distinct principal curvatures are constant, then $M^4$ is an isoparametric hypersurface, and the value of $S$ can only be $0, 4, 12$. This result supports Chern Conjecture.